Problems dealing with incomplete observations are especially frequent in survival analysis, where censored observations represent partial information which is valuable in estimating the distribution of survival times. Nonparametric Bayesian estimators for the distribution function have been proposed by Susarla and Van Ryzin. These estimators depend on the choice of a "Dirichlet process prior": Rai, Susarla, and Van Ryzin suggest an empirical Bayes approach to choosing the prior, resulting in estimators that "shrink" toward prior exponential survival curves. The research proposed herein will result in the definition and evaluation of a nonparametric Bayesian estimator which is a limit of estimators obtained using non-informative priors on finite sample spaces. This estimator differs from the Susarla-Van Ryzin estimators in that there is no need to choose from a family of priors: the priors used are uniquely determined. Thus this estimator should prove especially useful when no prior knowledge regarding the distribution of survival times is available. Also, the resulting estimator appears to make substantial use of the censored data. The major objectives of the proposed research are: (1) to obtain a tractable expression for the proposed estimator in terms of the observed data; (2) to investigate the properties of the proposed estimator; and (3) to compare the performance of this estimator with the Rai, Susarla, and Van Ryzin shrinkage estimators using computer simulations.